Summary
“The primary goal of this study is to investigate, using simulated data, the extent to which this ‘natural’ interpretation is robust and can be used to guide modelling decisions.”
- Period hyperparameter, \(P\), matches stellar rotation period.
- Timescale hyperparameter, \(\ell\), matches stellar spot evolution time.
- Harmonic complexity hyperparameter, \(\Gamma\), has weak relationship with physical parameters.
- No practical difference found between QP and QPC kernel results.
Quasi-periodic (QP) kernel
- Very common for stellar activity modelling
- Radial velocities (RVs) are believed to be quasi-periodic
- QP hyperparameters seem to have a natural interpretation
- rotation rates
- lifetimes of active regions
Squared Exponential kernel
\[k_\mathrm{SE}(\tau; A, \ell) = A \exp\left\{-\frac{1}{2}\left( \frac{\tau}{\ell}\right)^2\right\}\]
Periodic kernel
\[k_P(\tau; A, \Gamma, P) = A \exp\left\{ -\Gamma \sin^2\left[\pi \left( \frac{\tau}{P}\right) \right] \right\}\]
Cosine kernel
\[k_\mathrm{C}(\tau; A, P) = A \cos\left[2\pi \left( \frac{\tau}{P}\right)\right]\]
Quasi-periodic (QP) kernel
\[
k_\mathrm{QP}(\tau; A, \Gamma, P, \ell) = k_\mathrm{P}(\tau; A, \Gamma, P) \times k_\mathrm{SE}(\tau; A, \ell) = A \exp \left[ -\Gamma \sin^2 \left(\pi \frac{\tau}{P}\right) - \frac{\tau^2}{2\ell^2}\right]
\]
Quasi-periodic + Cosine (QPC) kernel
- Add a cosine term with a period equal to half of that of QP sine-squared term
- Capture the signal at the first harmonic of the stellar rotation period (Perger et al. 2021)
- a secondary peak in the autocorrelation function seen at lag equal to half the rotation period.
Quasi-periodic + Cosine (QPC) kernel
\[k_\mathrm{QPC}(\tau; A, \Gamma, P, \ell, f) = A \exp \left[- \frac{\tau^2}{2\ell^2}\right] \times \left( \exp \left[ -\Gamma \sin^2 \left(\pi \frac{\tau}{P}\right) \right] + f \cos\left( 4\pi\frac{\tau}{P}\right) \right)\]
Four Studies
- Degeneracy of QP hyperparameters
- Light Curve models
- Radial Velocity models
- Effect of Downsampling
Study 1: Hyperparameter Degeneracy
- Investigate the stability of Gaussian Process fitting when using QP or QPC kernels.
- Basically can you recover the ground truth from simulated data.
Study 1: Methodology
- Simulate light curves by drawing QP hyperparameters from priors.
- Fit a Gaussian Process using QP kernel.
- Check if the recovered hyperparameters match the one drawn from prior.
- Repeat for QPC kernel.
Study 1: Results
Study 1: K-S Test?
Studies 2 & 3: Methodology
- Simulate data from a model using known physical parameters.
- Fit a Gaussian Process with QP kernel.
- Compare estimated hyperparameters with known physical parameters.
- Repeat with QPC kernel.
- Study 2: Light curves
- Study 3: Radial velocities
Software Used
PySpot (Aigrain 2021) to simulate stellar spot light curve data.
george (Ambikasaran et al. 2015) for implementing kernels.
emcee (Foreman-Mackey et al. 2013) for MCMC..
Study 2: Model vs GP
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Study 4: Effect of Downsampling
- RV observations are typically ground-based and therefore more sparsely sampled and have higher noise.
- Methodology
- Add noise to simulated RV data.
- Randomly sample 50%, 20%, 10%, and 6% of the data.
- Visually examine the differences.
Study 4: Results
Study 4: Results
Takeaways for Fitting Stellar Models
- Shortest gap in data points \(\lt\) shortest rotation period.
- Rapidly rotating stars need large numbers of observations per season.
- Sparsity increases \(\rightarrow\) Timescale estimates decrease
- Undersampling means high frequency features cannot be resolved.
- Good results when \(\tau \gg\) sampling interval.
- Recovery of \(\Gamma\) is strongly affected by noise in data.
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